L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s − 3·11-s + 5·13-s + 3·15-s − 17-s − 2·19-s + 21-s − 6·23-s + 4·25-s − 27-s − 6·29-s + 4·31-s + 3·33-s + 3·35-s + 11·37-s − 5·39-s − 12·41-s + 43-s − 3·45-s − 12·47-s + 49-s + 51-s − 9·53-s + 9·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 0.774·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 0.507·35-s + 1.80·37-s − 0.800·39-s − 1.87·41-s + 0.152·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s + 0.140·51-s − 1.23·53-s + 1.21·55-s + ⋯ |
Λ(s)=(=(5712s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(5712s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.5209769691 |
L(21) |
≈ |
0.5209769691 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+T |
| 7 | 1+T |
| 17 | 1+T |
good | 5 | 1+3T+pT2 |
| 11 | 1+3T+pT2 |
| 13 | 1−5T+pT2 |
| 19 | 1+2T+pT2 |
| 23 | 1+6T+pT2 |
| 29 | 1+6T+pT2 |
| 31 | 1−4T+pT2 |
| 37 | 1−11T+pT2 |
| 41 | 1+12T+pT2 |
| 43 | 1−T+pT2 |
| 47 | 1+12T+pT2 |
| 53 | 1+9T+pT2 |
| 59 | 1−12T+pT2 |
| 61 | 1+10T+pT2 |
| 67 | 1+5T+pT2 |
| 71 | 1+pT2 |
| 73 | 1+7T+pT2 |
| 79 | 1−T+pT2 |
| 83 | 1−15T+pT2 |
| 89 | 1−9T+pT2 |
| 97 | 1+19T+pT2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.019507455750696322517359396577, −7.60071241079856865939505009051, −6.52974126119963944803026329454, −6.15891201432989065467850720102, −5.20983774777158442191812465551, −4.35865858694941866044297529040, −3.78371489917505067880314892179, −3.02402476613205357995053379893, −1.73527382204318293609960924592, −0.38832682507999745116601502102,
0.38832682507999745116601502102, 1.73527382204318293609960924592, 3.02402476613205357995053379893, 3.78371489917505067880314892179, 4.35865858694941866044297529040, 5.20983774777158442191812465551, 6.15891201432989065467850720102, 6.52974126119963944803026329454, 7.60071241079856865939505009051, 8.019507455750696322517359396577